Proof step 1 from proof propFOs-append (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  propFOs(t, l, bel0 + bel1) ≠ propFOs(t, l, bel0) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Proof step 2 from proof propFOs-append (previous step) (back to tree)

Heuristic structural induction applied rule structural induction

And obtained the following new goals

• Proof step 3:

  propFOs(t, l, (be + bel) + bel1) ≠ propFOs(t, l, be + bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Used simplifier rules: propFOs-base: ⊦ propFOs(t, l, []) = []; List('a) freely generated by [], +;; ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 3 from proof propFOs-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 4:

  propFOs(t, l, (be ' + bel) + bel1) ≠ propFOs(t, l, be ' + bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 4 from proof propFOs-append (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 5:

  be = SF, propFOs(t, l, (be ' + bel) + bel1) ≠ propFOs(t, l, be ' + bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

• Proof step 7:

  be = W(l0, v), propFOs(t, l, (be ' + bel) + bel1) ≠ propFOs(t, l, be ' + bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

• Proof step 9:

  be = FO(l0), propFOs(t, l, (be ' + bel) + bel1) ≠ propFOs(t, l, be ' + bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Used simplifier rules: Bevent freely generated by SF, W, FO;;

Proof step 5 from proof propFOs-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  propFOs(t, l, bel + bel1) ≠ propFOs(t, l, bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Used simplifier rules: propFOs-SF: ⊦ propFOs(t, l, SF ' + bel) = propFOs(t, l, bel); a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 6 from proof propFOs-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: t, l, bel1

Proof step 7 from proof propFOs-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  propFOs(t, l, bel + bel1) ≠ propFOs(t, l, bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Used simplifier rules: propFOs-W: ⊦ propFOs(t, l, W(l0, v) ' + bel) = propFOs(t, l, bel); a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 8 from proof propFOs-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: t, l, bel1

Proof step 9 from proof propFOs-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  propFOs(t, l, (FO(l0) ' + bel) + bel1) ≠ propFOs(t, l, FO(l0) ' + bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Proof step 10 from proof propFOs-append (previous step) (back to tree)

Heuristic cut applied rule cut formula

And obtained the following new goals

• Proof step 11:

  l = l0, propFOs(t, l, (FO(l0) ' + bel) + bel1) ≠ propFOs(t, l, FO(l0) ' + bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

• Proof step 13:

  l ≠ l0, propFOs(t, l, (FO(l0) ' + bel) + bel1) ≠ propFOs(t, l, FO(l0) ' + bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Used formulas: l = l0

Proof step 11 from proof propFOs-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  propFOs(t, l0, bel + bel1) ≠ propFOs(t, l0, bel) + propFOs(t, l0, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Used simplifier rules: propFOs-FO-same: ⊦ propFOs(t, l, FO(l) ' + bel) = FO(t) + propFOs(t, l, bel); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ a + x = a ' + x; (s)

Proof step 12 from proof propFOs-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: t, l0, bel1

Proof step 13 from proof propFOs-append (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  l ≠ l0, propFOs(t, l, bel + bel1) ≠ propFOs(t, l, bel) + propFOs(t, l, bel1), ∀ t, l, bel1. propFOs(t, l, bel + bel1) = propFOs(t, l, bel) + propFOs(t, l, bel1) ⊦

Rule argument

Additional information

Used simplifier rules: propFOs-FO-other: ⊦ l ≠ l0 → propFOs(t, l, FO(l0) ' + bel) = propFOs(t, l, bel); a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 14 from proof propFOs-append (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: t, l, bel1